# Nyquist Sampling Theorem. By: Arnold Evia

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Nyquist Sampling Theorem By: Arnold Evia

2 Table of Contents What is the Nyquist Sampling Theorem? Bandwidth Sampling Impulse Response Train Fourier Transform of Impulse Response Train Sampling in the Fourier Domain o Sampling cases Review

3 What is the Nyquist Sampling Theorem? Formal Definition: o If the frequency spectra of a function x(t) contains no frequencies higher than B hertz, x(t) is completely determined by giving its ordinates at a series of points spaced 1/(2B) seconds apart. In other words, to be able to accurately reconstruct a signal, samples must be recorded every 1/(2B) seconds, where B is the bandwidth of the signal.

4 Bandwidth There are many definitions to bandwidth depending on the application For signal processing, it is referred to as the range of frequencies above 0 ( F(w) of f(t)) Signals that have a definite value for the highest frequency are bandlimited ( F(w) =0 for w >B) In reality, signals are never bandlimited o In order to be bandlimited, the signal must have infinite duration in time Bandlimited signal with bandwidth B Non-bandlimited signal (representative of real signals)

5 Sampling Sampling is recording values of a function at certain times Allows for transformation of a continuous time function to a discrete time function This is obtained by multiplication of f(t) by a unit impulse train

6 Impulse Response Train Consider an impulse train: Sometimes referred to as comb function Periodic with a value of 1 for every nt0, where n is integer values from - to, and 0 elsewhere

7 Fourier Transform of Impulse Train Input Set up the Equations function into the fourier transform eqs. T0 is the period of the func. Solve Solve Dn for one Dn period Consider period from T0/2 to T0/2 Only one value: at t=0 Integral equates to 1 as e-jnw0(0) = 1 Substitute Understand Dn into Answer first equation The fourier spectra of the function has an amplitude of 1/T0 at nw0 for values of n from to +, and 0 elsewhere Distance between each w0 is dependent on T0. Decreasing T0, increases the w0 and distance Original Function Fourier Spectra

8 Sampling in the Fourier Domain Consider a bandlimited signal f(t) multiplied with an impulse response train (sampled): o If the period of the impulse train is insufficient (T0 > 1/(2B)), aliasing occurs o When T0=1/(2B), T0 is considered the nyquist rate. 1/T0 is the nyquist frequency Recall that multiplication in the time domain is convolution. in the frequency domain: = As can be seen in the fourier spectra, it is isual Representation of Property ime omain req. omain only necessary to extract the fourier spectra from one period to reconstruct the signal! * =

9 Sampling Cases T0>1/(2B) o Undersampling o Distance between copies of F(w) that overlap happens o Aliasing occurs, and the higher frequencies of the signal are corrupted T0<=1/(2B) o Oversampling o Distance between copies of F(w) is sufficient enough to prevent overlap o Spectra can be filtered to accurately reconstruct signal

10 Review Nyquist sampling rate is the rate which samples of the signal must be recorded in order to accurately reconstruct the sampled signal o Must satisfy T0 <= 1/(2B); where T0 is the time between recorded samples and B is the bandwidth of the signal A signal sampled every T0 seconds can be represented as: where Ts = T0

11 Review (cont.) One way of understanding the importance of the Nyquist sampling rate is observing the fourier spectra of a sampled signal A sampled signal s fourier spectra is a periodic function of the original unsampled signal s fourier spectra o Therefore, it is only necessary to extract the data from one period to accurately reconstruct the signal Aliasing can occur if the sampling rate is less than the Nyquist sampling rate o There is overlap in the fourier spectra, and the signal cannot be accurately reconstructed (Undersampling)

12 References Some basic resources can be found here: ARC website: ARC BME schedule: html

### SGN-1158 Introduction to Signal Processing Test. Solutions

SGN-1158 Introduction to Signal Processing Test. Solutions 1. Convolve the function ( ) with itself and show that the Fourier transform of the result is the square of the Fourier transform of ( ). (Hints:

### Sampling Theorem Notes. Recall: That a time sampled signal is like taking a snap shot or picture of signal periodically.

Sampling Theorem We will show that a band limited signal can be reconstructed exactly from its discrete time samples. Recall: That a time sampled signal is like taking a snap shot or picture of signal

### Continuous time vs Discrete time. Lecture 13. Sampling & Discrete signals (Lathi ) Sampling Process. Sampling Theorem. f s. = 500Hz ELECTRONICS

Lecture 13 Sampling & Discrete signals (Lathi 8.1-8.2) Peter Cheung Department of Electrical & Electronic Engineering Imperial College London URL: www.ee.imperial.ac.uk/pcheung/teaching/ee2_signals E-mail:

### Computer Graphics. Course SS 2007 Antialiasing. computer graphics & visualization

Computer Graphics Course SS 2007 Antialiasing How to avoid spatial aliasing caused by an undersampling of the signal, i.e. the sampling frequency is not high enough to cover all details Supersampling -

### Basics on Digital Signal Processing

Basics on Digital Signal Processing Introduction Vassilis Anastassopoulos Electronics Laboratory, Physics Department, University of Patras Outline of the Course 1. Introduction (sampling quantization)

### L6: Short-time Fourier analysis and synthesis

L6: Short-time Fourier analysis and synthesis Overview Analysis: Fourier-transform view Analysis: filtering view Synthesis: filter bank summation (FBS) method Synthesis: overlap-add (OLA) method STFT magnitude

### Lecture 2: 2D Fourier transforms and applications

Lecture 2: 2D Fourier transforms and applications B14 Image Analysis Michaelmas 2014 A. Zisserman Fourier transforms and spatial frequencies in 2D Definition and meaning The Convolution Theorem Applications

### Aliasing, Image Sampling and Reconstruction

Aliasing, Image Sampling and Reconstruction Recall: a pixel is a point It is NOT a box, disc or teeny wee light It has no dimension It occupies no area It can have a coordinate More than a point, it is

### Signaling is the way data is communicated. This type of signal used can be either analog or digital

3.1 Analog vs. Digital Signaling is the way data is communicated. This type of signal used can be either analog or digital 1 3.1 Analog vs. Digital 2 WCB/McGraw-Hill The McGraw-Hill Companies, Inc., 1998

### UNIVERSITY OF LONDON GOLDSMITHS COLLEGE. B. Sc. Examination Creative Computing. IS52020A (CC227) Creative Computing 2.

UNIVERSITY OF LONDON GOLDSMITHS COLLEGE B. Sc. Examination 2011 Creative Computing IS52020A (CC227) Creative Computing 2 Duration: 3 hours Date and time: There are six questions in this paper. You should

Measurement Lab Fourier Analysis Last Modified 9/5/06 Any time-varying signal can be constructed by adding together sine waves of appropriate frequency, amplitude, and phase. Fourier analysis is a technique

### CSE168 Computer Graphics II, Rendering. Spring 2006 Matthias Zwicker

CSE168 Computer Graphics II, Rendering Spring 2006 Matthias Zwicker Last time Sampling and aliasing Aliasing Moire patterns Aliasing Sufficiently sampled Insufficiently sampled [R. Cook ] Fourier analysis

### 5 Signal Design for Bandlimited Channels

225 5 Signal Design for Bandlimited Channels So far, we have not imposed any bandwidth constraints on the transmitted passband signal, or equivalently, on the transmitted baseband signal s b (t) I[k]g

### Analysis/resynthesis with the short time Fourier transform

Analysis/resynthesis with the short time Fourier transform summer 2006 lecture on analysis, modeling and transformation of audio signals Axel Röbel Institute of communication science TU-Berlin IRCAM Analysis/Synthesis

### Moving Average Filters

CHAPTER 15 Moving Average Filters The moving average is the most common filter in DSP, mainly because it is the easiest digital filter to understand and use. In spite of its simplicity, the moving average

### Purpose of Time Series Analysis. Autocovariance Function. Autocorrelation Function. Part 3: Time Series I

Part 3: Time Series I Purpose of Time Series Analysis (Figure from Panofsky and Brier 1968) Autocorrelation Function Harmonic Analysis Spectrum Analysis Data Window Significance Tests Some major purposes

### Sampling and Interpolation. Yao Wang Polytechnic University, Brooklyn, NY11201

Sampling and Interpolation Yao Wang Polytechnic University, Brooklyn, NY1121 http://eeweb.poly.edu/~yao Outline Basics of sampling and quantization A/D and D/A converters Sampling Nyquist sampling theorem

### SIGNAL PROCESSING & SIMULATION NEWSLETTER

1 of 10 1/25/2008 3:38 AM SIGNAL PROCESSING & SIMULATION NEWSLETTER Note: This is not a particularly interesting topic for anyone other than those who ar e involved in simulation. So if you have difficulty

### 4.3 Analog-to-Digital Conversion

4.3 Analog-to-Digital Conversion overview including timing considerations block diagram of a device using a DAC and comparator example of a digitized spectrum number of data points required to describe

### What is a Filter? Output Signal. Input Signal Amplitude. Frequency. Low Pass Filter

What is a Filter? Input Signal Amplitude Output Signal Frequency Time Sequence Low Pass Filter Time Sequence What is a Filter Input Signal Amplitude Output Signal Frequency Signal Noise Signal Noise Frequency

### The continuous and discrete Fourier transforms

FYSA21 Mathematical Tools in Science The continuous and discrete Fourier transforms Lennart Lindegren Lund Observatory (Department of Astronomy, Lund University) 1 The continuous Fourier transform 1.1

### Fast Fourier Transforms and Power Spectra in LabVIEW

Application Note 4 Introduction Fast Fourier Transforms and Power Spectra in LabVIEW K. Fahy, E. Pérez Ph.D. The Fourier transform is one of the most powerful signal analysis tools, applicable to a wide

### Time series analysis Matlab tutorial. Joachim Gross

Time series analysis Matlab tutorial Joachim Gross Outline Terminology Sampling theorem Plotting Baseline correction Detrending Smoothing Filtering Decimation Remarks Focus on practical aspects, exercises,

### PYKC Jan-7-10. Lecture 1 Slide 1

Aims and Objectives E 2.5 Signals & Linear Systems Peter Cheung Department of Electrical & Electronic Engineering Imperial College London! By the end of the course, you would have understood: Basic signal

### 7. FOURIER ANALYSIS AND DATA PROCESSING

7. FOURIER ANALYSIS AND DATA PROCESSING Fourier 1 analysis plays a dominant role in the treatment of vibrations of mechanical systems responding to deterministic or stochastic excitation, and, as has already

### Discrete Fourier Series & Discrete Fourier Transform Chapter Intended Learning Outcomes

Discrete Fourier Series & Discrete Fourier Transform Chapter Intended Learning Outcomes (i) Understanding the relationships between the transform, discrete-time Fourier transform (DTFT), discrete Fourier

### Fourier Transform and Image Filtering. CS/BIOEN 6640 Lecture Marcel Prastawa Fall 2010

Fourier Transform and Image Filtering CS/BIOEN 6640 Lecture Marcel Prastawa Fall 2010 The Fourier Transform Fourier Transform Forward, mapping to frequency domain: Backward, inverse mapping to time domain:

### Fourier Transform and Convolution

CHAPTER 3 Fourier Transform and Convolution 3.1 INTRODUCTION In this chapter, both the Fourier transform and Fourier series will be discussed. The properties of the Fourier transform will be presented

### Voice---is analog in character and moves in the form of waves. 3-important wave-characteristics:

Voice Transmission --Basic Concepts-- Voice---is analog in character and moves in the form of waves. 3-important wave-characteristics: Amplitude Frequency Phase Voice Digitization in the POTS Traditional

### Lab 4 Sampling, Aliasing, FIR Filtering

47 Lab 4 Sampling, Aliasing, FIR Filtering This is a software lab. In your report, please include all Matlab code, numerical results, plots, and your explanations of the theoretical questions. The due

### Computer Networks and Internets, 5e Chapter 6 Information Sources and Signals. Introduction

Computer Networks and Internets, 5e Chapter 6 Information Sources and Signals Modified from the lecture slides of Lami Kaya (LKaya@ieee.org) for use CECS 474, Fall 2008. 2009 Pearson Education Inc., Upper

### A Tutorial on Fourier Analysis

A Tutorial on Fourier Analysis Douglas Eck University of Montreal NYU March 26 1.5 A fundamental and three odd harmonics (3,5,7) fund (freq 1) 3rd harm 5th harm 7th harmm.5 1 2 4 6 8 1 12 14 16 18 2 1.5

### where T o defines the period of the signal. The period is related to the signal frequency according to

SIGNAL SPECTRA AND EMC One of the critical aspects of sound EMC analysis and design is a basic understanding of signal spectra. A knowledge of the approximate spectral content of common signal types provides

### T = 1 f. Phase. Measure of relative position in time within a single period of a signal For a periodic signal f(t), phase is fractional part t p

Data Transmission Concepts and terminology Transmission terminology Transmission from transmitter to receiver goes over some transmission medium using electromagnetic waves Guided media. Waves are guided

### Lecture 8 ELE 301: Signals and Systems

Lecture 8 ELE 3: Signals and Systems Prof. Paul Cuff Princeton University Fall 2-2 Cuff (Lecture 7) ELE 3: Signals and Systems Fall 2-2 / 37 Properties of the Fourier Transform Properties of the Fourier

### Topic 4: Continuous-Time Fourier Transform (CTFT)

ELEC264: Signals And Systems Topic 4: Continuous-Time Fourier Transform (CTFT) Aishy Amer Concordia University Electrical and Computer Engineering o Introduction to Fourier Transform o Fourier transform

Chapter 7 Signals and Sampling 7.1 Sampling Theory 7.2 Image Sampling Interface Chapter 7 of Physically Based Rendering by Pharr&Humphreys 7.3 Stratified Sampling 7.4 Low-Discrepancy Sampling 7.5 Best-Candidate

### Lecture 9, Multirate Signal Processing, z-domain Effects

Lecture 9, Multirate Signal Processing, z-domain Effects Last time we saw the effects of downsampling and upsampling. Observe that we can perfectly reconstruct the high pass signal in our example if we

### Continuous Time Signals (Part - I) Fourier series

Continuous Time Signals (Part - I) Fourier series (a) Basics 1. Which of the following signals is/are periodic? (a) s(t) = cos t + cos 3t + cos 5t (b) s(t) = exp(j8 πt) (c) s(t) = exp( 7t) sin 1πt (d)

### Probability and Random Variables. Generation of random variables (r.v.)

Probability and Random Variables Method for generating random variables with a specified probability distribution function. Gaussian And Markov Processes Characterization of Stationary Random Process Linearly

### Chapter 3 Discrete-Time Fourier Series. by the French mathematician Jean Baptiste Joseph Fourier in the early 1800 s. The

Chapter 3 Discrete-Time Fourier Series 3.1 Introduction The Fourier series and Fourier transforms are mathematical correlations between the time and frequency domains. They are the result of the heat-transfer

### Frequency Response of FIR Filters

Frequency Response of FIR Filters Chapter 6 This chapter continues the study of FIR filters from Chapter 5, but the emphasis is frequency response, which relates to how the filter responds to an input

### Correlation and Convolution Class Notes for CMSC 426, Fall 2005 David Jacobs

Correlation and Convolution Class otes for CMSC 46, Fall 5 David Jacobs Introduction Correlation and Convolution are basic operations that we will perform to extract information from images. They are in

### Image Enhancement in the Frequency Domain

Image Enhancement in the Frequency Domain Jesus J. Caban Outline! Assignment #! Paper Presentation & Schedule! Frequency Domain! Mathematical Morphology %& Assignment #! Questions?! How s OpenCV?! You

### CONVOLUTION Digital Signal Processing

CONVOLUTION Digital Signal Processing Introduction As digital signal processing continues to emerge as a major discipline in the field of electrical engineering an even greater demand has evolved to understand

### Analog and Digital Signals, Time and Frequency Representation of Signals

1 Analog and Digital Signals, Time and Frequency Representation of Signals Required reading: Garcia 3.1, 3.2 CSE 3213, Fall 2010 Instructor: N. Vlajic 2 Data vs. Signal Analog vs. Digital Analog Signals

### Image Formation. Image Formation occurs when a sensor registers radiation. Mathematical models of image formation:

Image Formation Image Formation occurs when a sensor registers radiation. Mathematical models of image formation: 1. Image function model 2. Geometrical model 3. Radiometrical model 4. Color model 5. Spatial

### FIR Filter Design. FIR Filters and the z-domain. The z-domain model of a general FIR filter is shown in Figure 1. Figure 1

FIR Filters and the -Domain FIR Filter Design The -domain model of a general FIR filter is shown in Figure. Figure Each - box indicates a further delay of one sampling period. For example, the input to

### Chapter 6. Power Spectrum. 6.1 Outline

Chapter 6 Power Spectrum The power spectrum answers the question How much of the signal is at a frequency ω?. We have seen that periodic signals give peaks at a fundamental and its harmonics; quasiperiodic

### EECS 556 Image Processing W 09. Interpolation. Interpolation techniques B splines

EECS 556 Image Processing W 09 Interpolation Interpolation techniques B splines What is image processing? Image processing is the application of 2D signal processing methods to images Image representation

### Analog Representations of Sound

Analog Representations of Sound Magnified phonograph grooves, viewed from above: The shape of the grooves encodes the continuously varying audio signal. Analog to Digital Recording Chain ADC Microphone

### Digital Transmission (Line Coding)

Digital Transmission (Line Coding) Pulse Transmission Source Multiplexer Line Coder Line Coding: Output of the multiplexer (TDM) is coded into electrical pulses or waveforms for the purpose of transmission

### Antialiasing. CS 319 Advanced Topics in Computer Graphics John C. Hart

Antialiasing CS 319 Advanced Topics in Computer Graphics John C. Hart Aliasing Aliasing occurs when signals are sampled too infrequently, giving the illusion of a lower frequency signal alias noun (c.

### Fourier Analysis. Nikki Truss. Abstract:

Fourier Analysis Nikki Truss 09369481 Abstract: The aim of this experiment was to investigate the Fourier transforms of periodic waveforms, and using harmonic analysis of Fourier transforms to gain information

### Difference Equations

Difference Equations Andrew W H House 10 June 004 1 The Basics of Difference Equations Recall that in a previous section we saw that IIR systems cannot be evaluated using the convolution sum because it

### PYKC 3-Mar-11. Lecture 15 Slide 1. Laplace transform. Fourier transform. Discrete Fourier transform. transform. L5.8 p560.

- derived from Laplace Lecture 15 Discrete-Time System Analysis using -Transform (Lathi 5.1) Peter Cheung Department of Electrical & Electronic Engineering Imperial College London URL: www.ee.imperial.ac.uk/pcheung/teaching/ee2_signals

### The Complex Fourier Transform

CHAPTER 31 The Complex Fourier Transform Although complex numbers are fundamentally disconnected from our reality, they can be used to solve science and engineering problems in two ways. First, the parameters

### Convolution, Correlation, & Fourier Transforms. James R. Graham 10/25/2005

Convolution, Correlation, & Fourier Transforms James R. Graham 10/25/2005 Introduction A large class of signal processing techniques fall under the category of Fourier transform methods These methods fall

### Fourier Series. Some Properties of Functions. Philippe B. Laval. Today KSU. Philippe B. Laval (KSU) Fourier Series Today 1 / 19

Fourier Series Some Properties of Functions Philippe B. Laval KSU Today Philippe B. Laval (KSU) Fourier Series Today 1 / 19 Introduction We review some results about functions which play an important role

### Sampling Theory For Digital Audio By Dan Lavry, Lavry Engineering, Inc.

Sampling Theory Page Copyright Dan Lavry, Lavry Engineering, Inc, 24 Sampling Theory For Digital Audio By Dan Lavry, Lavry Engineering, Inc. Credit: Dr. Nyquist discovered the sampling theorem, one of

### Implementation of Digital Signal Processing: Some Background on GFSK Modulation

Implementation of Digital Signal Processing: Some Background on GFSK Modulation Sabih H. Gerez University of Twente, Department of Electrical Engineering s.h.gerez@utwente.nl Version 4 (February 7, 2013)

### Voice Signal s Noise Reduction Using Adaptive/Reconfigurable Filters for the Command of an Industrial Robot

Voice Signal s Noise Reduction Using Adaptive/Reconfigurable Filters for the Command of an Industrial Robot Moisa Claudia**, Silaghi Helga Maria**, Rohde L. Ulrich * ***, Silaghi Paul****, Silaghi Andrei****

### 07 SAMPLING AND RECONSTRUCTION

07 SAMPLING AND RECONSTRUCTION Although the final output of a renderer like pbrt is a two-dimensional grid of colored pixels, incident radiance is actually a continuous function defined over the film plane.

### R U S S E L L L. H E R M A N

R U S S E L L L. H E R M A N A N I N T R O D U C T I O N T O F O U R I E R A N D C O M P L E X A N A LY S I S W I T H A P P L I C AT I O N S T O T H E S P E C T R A L A N A LY S I S O F S I G N A L S R.

### BSEE Degree Plan Bachelor of Science in Electrical Engineering: 2015-16

BSEE Degree Plan Bachelor of Science in Electrical Engineering: 2015-16 Freshman Year ENG 1003 Composition I 3 ENG 1013 Composition II 3 ENGR 1402 Concepts of Engineering 2 PHYS 2034 University Physics

### SHOCK AND VIBRATION RESPONSE SPECTRA COURSE Unit 18. Filtering

SHOCK AND VIBRATION RESPONSE SPECTRA COURSE Unit 18. Filtering By Tom Irvine Email: tomirvine@aol.com Introduction Filtering is a tool for resolving signals. Filtering can be performed on either analog

### NRZ Bandwidth - HF Cutoff vs. SNR

Application Note: HFAN-09.0. Rev.2; 04/08 NRZ Bandwidth - HF Cutoff vs. SNR Functional Diagrams Pin Configurations appear at end of data sheet. Functional Diagrams continued at end of data sheet. UCSP

### Linear Filtering Part II

Linear Filtering Part II Selim Aksoy Department of Computer Engineering Bilkent University saksoy@cs.bilkent.edu.tr Fourier theory Jean Baptiste Joseph Fourier had a crazy idea: Any periodic function can

### The University of Texas at Austin Dept. of Electrical and Computer Engineering Midterm #1. Spirou et Fantasio

The University of Texas at Austin Dept. of Electrical and Computer Engineering Midterm #1 Date: October 14, 2016 Course: EE 445S Evans Name: Spirou et Fantasio Last, First The exam is scheduled to last

### Improvement of The ADC Resolution Based on FPGA Implementation of Interpolating Algorithm

International Journal of New Technology and Research (IJNTR) ISSN:2454-4116, Volume-2, Issue-1, January 2016 Pages 100-103 Improvement of The ADC Resolution Based on FPGA Implementation of Interpolating

### OSE801 Engineering System Identification Fall 2012 Lecture 5: Fourier Analysis: Introduction

OSE801 Engineering System Identification Fall 2012 Lecture 5: Fourier Analysis: Introduction Instructors: K. C. Park and I. K. Oh (Division of Ocean Systems Engineering) System-Identified State Space Model

### The Fourier Analysis Tool in Microsoft Excel

The Fourier Analysis Tool in Microsoft Excel Douglas A. Kerr Issue March 4, 2009 ABSTRACT AD ITRODUCTIO The spreadsheet application Microsoft Excel includes a tool that will calculate the discrete Fourier

### Short-time FFT, Multi-taper analysis & Filtering in SPM12

Short-time FFT, Multi-taper analysis & Filtering in SPM12 Computational Psychiatry Seminar, FS 2015 Daniel Renz, Translational Neuromodeling Unit, ETHZ & UZH 20.03.2015 Overview Refresher Short-time Fourier

### Agilent Creating Multi-tone Signals With the N7509A Waveform Generation Toolbox. Application Note

Agilent Creating Multi-tone Signals With the N7509A Waveform Generation Toolbox Application Note Introduction Of all the signal engines in the N7509A, the most complex is the multi-tone engine. This application

### Analog Measurand: Time Dependant Characteristics

Chapter 4 Analog Measurand: Time Dependant Characteristics 4.1 Introduction A parameter common to all of measurement is time: All measurands have time-related characteristics. As time progresses, the magnitude

### Signals and Sampling. CMPT 461/761 Image Synthesis Torsten Möller. Machiraju/Möller

Signals and Sampling CMPT 461/761 Image Synthesis Torsten Möller Reading Chapter 7 of Physically Based Rendering by Pharr&Humphreys Chapter 14.10 of CG: Principles & Practice by Foley, van Dam et al. Chapter

### 1995 Mixed-Signal Products SLAA013

Application Report 995 Mixed-Signal Products SLAA03 IMPORTANT NOTICE Texas Instruments and its subsidiaries (TI) reserve the right to make changes to their products or to discontinue any product or service

### EE 179 April 21, 2014 Digital and Analog Communication Systems Handout #16 Homework #2 Solutions

EE 79 April, 04 Digital and Analog Communication Systems Handout #6 Homework # Solutions. Operations on signals (Lathi& Ding.3-3). For the signal g(t) shown below, sketch: a. g(t 4); b. g(t/.5); c. g(t

### Department of Electronics and Communication Engineering 1

DHANALAKSHMI COLLEGE OF ENGINEERING, CHENNAI DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING III Year ECE / V Semester EC 6502 PRINCIPLES OF DIGITAL SIGNAL PROCESSING QUESTION BANK Department of

### Difference Equations and Digital Filters

SIGNALS AND SYSTEMS: PAPER C HANDOUT 8. Dr David Corrigan. Electronic and Electrical Engineering Dept. corrigad@tcd.ie www.mee.tcd.ie/ corrigad Difference Equations and Digital Filters The last topic discussed

### EE 179 April 28, 2014 Digital and Analog Communication Systems Handout #21 Homework #3 Solutions

EE 179 April 28, 214 Digital and Analog Communication Systems Handout #21 Homework #3 Solutions 1. DSB-SC modulator (Lathi& Ding 4.2-3). You are asked to design a DSB-SC modulator to generate a modulated

### 2 Background: Fourier Series Analysis and Synthesis

GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL and COMPUTER ENGINEERING ECE 2025 Spring 2001 Lab #11: Design with Fourier Series Date: 3 6 April 2001 This is the official Lab #11 description. The

### The Fourier Transform

The Fourier Transorm Fourier Series Fourier Transorm The Basic Theorems and Applications Sampling Bracewell, R. The Fourier Transorm and Its Applications, 3rd ed. New York: McGraw-Hill, 2. Eric W. Weisstein.

### Elec 484 Final Project Report. Marlon Smith

Elec 484 Final Project Report Marlon Smith Abstract This report discusses the implementation of a variety of audio effects using a phase vocoder. Effects such as time stretching, pitch shifting, and robotization

### EECS 206 Solutions to Midterm Exam 2 July 12, Problems 1 to 10 are multiple-choice. Each has 7 points. No partial credit will be given.

EECS 06 Solutions to Midterm Exam July, 00 Instructions: Answer on this questionnaire Print your name Sign the pledge below Closed book and notes One 8 / x sheet of paper allowed Calculators allowed Read

### chapter Introduction to Digital Signal Processing and Digital Filtering 1.1 Introduction 1.2 Historical Perspective

Introduction to Digital Signal Processing and Digital Filtering chapter 1 Introduction to Digital Signal Processing and Digital Filtering 1.1 Introduction Digital signal processing (DSP) refers to anything

Performance of an IF sampling ADC in receiver applications David Buchanan Staff Applications Engineer Analog Devices, Inc. Introduction The concept of direct intermediate frequency (IF) sampling is not

### Waves. Wave Parameters. Krauss Chapter Nine

Waves Krauss Chapter Nine Wave Parameters Wavelength = λ = Length between wave crests (or troughs) Wave Number = κ = 2π/λ (units of 1/length) Wave Period = T = Time it takes a wave crest to travel one

### Applications of Fourier transform. Convolution. Notes. Notes. Notes. Notes

Applications of Fourier transform So far, only considered Fourier transform as a way to obtain the frequency spectrum of a function/signal. However, there are other important applications: : Real physical

### Frequency-Domain Analysis: the Discrete Fourier Series and the Fourier Transform

Frequency-Domain Analysis: the Discrete Fourier Series and the Fourier Transform John Chiverton School of Information Technology Mae Fah Luang University 1st Semester 2009/ 2552 Outline Overview Lecture

### Motorola Digital Signal Processors

Motorola Digital Signal Processors Principles of Sigma-Delta Modulation for Analog-to- Digital Converters by Sangil Park, Ph. D. Strategic Applications Digital Signal Processor Operation MOTOROLA APR8

### AUDIO SIGNAL EXTRAPOLATION - THEORY AND APPLICATIONS. Ismo Kauppinen and Kari Roth

Proc. of the 5 th Int. Conference on Digital Audio Effects (DAFx-), Hamburg, Germany, September 6-8, AUDIO SIGNAL EXTRAPOLATION - THEORY AND APPLICATIONS Ismo Kauppinen and Kari Roth University of Turku

### TCOM 370 NOTES 99-2 FOURIER SERIES, BANDWIDTH, AND SIGNALING RATES ON DATA TRANSMISSION LINKS

TCOM 37 NOTES 99-2 FOURIER SERIES, BANDWIDTH, AND SIGNALING RATES ON DATA TRANSMISSION LINKS. PULSE TRANSMISSION We now consider some fundamental aspects of binary data transmission over an individual

### Seismic data interpolation using a fast generalized Fourier transform

Seismic data interpolation using a fast generalized Fourier transform Mostafa Naghizadeh and Kris Innanen CREWES University of Calgary CREWES annual sponsor s meeting Banff, Alberta 2 December 2010 Outlines:

### SAMPLE: EXPERIMENT 10 Bandpass Filter / Fourier Analysis

SAMPLE: EXPERIMENT 10 Bandpass Filter / Fourier Analysis ---------------------------------------------------------------------------------------------------- This experiment is an excerpt from: Electric

### A Brief Introduction to Sigma Delta Conversion

A Brief Introduction to Sigma Delta Conversion Application Note May 995 AN954 Author: David Jarman Introduction The sigma delta conversion techniue has been in existence for many years, but recent technological

### IMPLEMENTATION OF FIR FILTER USING EFFICIENT WINDOW FUNCTION AND ITS APPLICATION IN FILTERING A SPEECH SIGNAL

IMPLEMENTATION OF FIR FILTER USING EFFICIENT WINDOW FUNCTION AND ITS APPLICATION IN FILTERING A SPEECH SIGNAL Saurabh Singh Rajput, Dr.S.S. Bhadauria Department of Electronics, Madhav Institute of Technology

### Introduction of Fourier Analysis and Time-frequency Analysis

Introduction of Fourier Analysis and Time-frequency Analysis March 1, 2016 Fourier Series Fourier transform Fourier analysis Mathematics compares the most diverse phenomena and discovers the secret analogies